A bead-mass oscillatory system problem

[Zayan]

Homework Statement

A bead of mass m can slide on a smooth straight horizontal wire and a particle of mass 2m is attached to the bead by a light string of length l. Initially, the particle is held in contact with the wire and the string is just taut. Then, the particle is released to fall under the gravity.
Find the tension when the string makes 37 degrees with the horizontal.

Relevant Equations

W=∆K.E
m1v1=m2v2
a= mv²/r

I can't figure out how to find the velocity of the particle at 37 degrees. Basically the bead moves with velocity towards right let's call it v1. The particle moves with some velocity v2. In frame of the bead, the particle is performing circular motion. So v of particle wrt bead would be perpendicular to the string. But how would I find the velocity of particle in ground frame?

I tried using vectors to figure it out and the angle is coming out to be extremely long. One equation is by work energy theorem, the second would be by momentum conservation in horizontal direction but for that I require the ground frame velocity. After finding the velocity I think I am supposed to write the equation of motion of particle wrt bead as it is performing circular motion wrt to it.

 

[wrobel]

Introduce an angle $\alpha$ between the string and the wire and a coordinate x that determines the position of the bead.
Express in these terms the equation of energy conservation and the equation of momentum conservation in the horizontal direction. Find the constants of these first integrals by using the initial conditions.
Differentiating these equations in t you can obtain the equations of system's motion.
Then use 2nd Newton to find string's tension.

 

[Zayan]

Introduce an angle $\alpha$ between the string and the wire and a coordinate x

With respect to what do I write the coordinates? The bead is moving itself and if I wanted to write the coordinates wrt let's say, the left end, I would require the velocity of bead at all time.

 

[wrobel]

The equations mentioned above contain the terms $\alpha,\dot\alpha,\dot x$

 

[Zayan]

The equations mentioned above contain the terms $\alpha,\dot\alpha,\dot x$

How do I conserve the energy. There is also vertical velocity so shouldn't dy/dt also be taken into account